Three-time-scale nonlinear control of an autonomous helicopter on a platform

Resumen

A three-time scale singular perturbation control is applied to an autonomous helicopter model on a platform to regulate its vertical position. Two singularly perturbation time-scale analysis approaches are presented, the Top-Down (TD), and the Bottom-Up (BU), which permit to analyze multi-time scale systems. These methodologies are based in a sequential application of the general two-time-scale singular perturbation formulation, allowing to decouple the helicopter three-time-scale problem into two simpler two-time-scale models. The TD and BU methodologies provide a step-by-step procedure that allows to design the proper control laws that allows to achieve the desired helicopter�s altitude by either actuating on both the collective pitch angle and the angular velocity of the blades. In addition, the same methodology, provides a tool to select an appropriate composite Lyapunov function for the complete singularly perturbed system, and to demonstrate the asymptotic stability for the resulting closed-loop nonlinear singularly perturbed system for sufficiently small singular perturbation parameters using Lyapunov stability methods, and everything in an all-in-one step-by-step methodology. The equivalency between both the TD and BU methodologies, permits the designer to choose which direction is to be used, depending on the structure of the system to be studied, and in special cases, determine which combination of both methodologies is the most appropriate according to the natural flow of the variables. The validity of the methodology has been proved by obtaining the stability upper bound limits for the three-time-scale boundary layers ensuring that the selected parasitic constants for the proposed control law satisfy the bounds for both the helicopter and the simplified model here employed. The stability results have also presented a closed form solution for the proper selection of the stability parameters such that fulfill the required growth requirements among different singularly perturbed subsystem, providing asymptotic stability for the helicopter full system with prescribed upper bounds on the parasitic parameters. The TD and BU time scale analysis is also extended to the more general Nth-time-scale analysis using a 4th-time-scale general example. The sequential strategy of decomposing the 4th-time-scale system, into simpler two-time-scale subsystems provides valuable tools for both the analysis of time-scale singularlyperturbed systems, and the stability properties of any general singularly perturbed Nth-time-scale system.